AMC 8 · 2005 · #2

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentagemulti-digit-arithmeticfraction-multiplication identify-subproblems ↑ Prerequisites: multi-digit-arithmeticpercentage

📏 Short solution 💡 2 insights

Problem

Karl bought five folders from Pay-A-Lot at a cost of $\textdollar 2.50$ each.
Pay-A-Lot had a 20%-off sale the following day. How much could
Karl have saved on the purchase by waiting a day?

Pick an answer.

(A)

$\textdollar 1.00$

(B)

$\textdollar 2.00$

(C)

$\textdollar 2.50$

(D)

$\textdollar 2.75$

(E)

$\textdollar 5.00$

AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: Karl bought $5$ folders at $\$2.50$ each. The next day, the store had a $20\%$-off sale. How much money could Karl have saved by waiting one day?

Givens: Number of folders bought $= 5$; Price per folder $= \$2.50$; Next-day sale $= 20\%$ off; Answer choices: (A) $\$1.00$, (B) $\$2.00$, (C) $\$2.50$, (D) $\$2.75$, (E) $\$5.00$

Unknowns: The savings (in dollars) Karl would have made by buying during the sale

Understand

Restated: Karl bought $5$ folders at $\$2.50$ each. The next day, the store had a $20\%$-off sale. How much money could Karl have saved by waiting one day?

Givens: Number of folders bought $= 5$; Price per folder $= \$2.50$; Next-day sale $= 20\%$ off; Answer choices: (A) $\$1.00$, (B) $\$2.00$, (C) $\$2.50$, (D) $\$2.75$, (E) $\$5.00$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #16 Change Focus

The question splits cleanly into two small jobs (Tool #7, Identify Subproblems): first find the original total cost of all $5$ folders, then take $20\%$ of that total to get the savings. Tool #16 (Change Focus) gives a faster shortcut: since $20\% = \tfrac{1}{5}$, taking $20\%$ of "$5$ folders" is the same as the cost of $1$ folder. Both routes land on the same answer; the subproblem route confirms it with a direct calculation.

Execute — Answer: C

#7 Identify Subproblems 5.NBT.B.7 Step 1

First subproblem: find the original total cost.
Five folders at $\$2.50$ each is a multiplication.

$\text{Total cost} = 5 \times \$2.50 = \$12.50$

💡 Multiplying a count by a per-item price is the Grade 5 decimal multiplication move.

#7 Identify Subproblems 6.RP.A.3 Step 2

Second subproblem: find $20\%$ of the total cost.
Convert $20\%$ to the fraction $\tfrac{1}{5}$ or the decimal $0.20$ and multiply by the total.

$\text{Savings} = 20\% \times \$12.50 = \dfrac{1}{5} \times \$12.50 = \$2.50 \;\Rightarrow\; \textbf{(C)}$

💡 "Percent of a quantity" is the Grade 6 percent-as-rate-per-100 calculation: $20\%$ of $\$12.50$ means $\tfrac{20}{100} \times \$12.50$.

[1] #7 5.NBT.B.7 First subproblem: find the original total cost. Five folders at $\$2.50$ each is

[2] #7 6.RP.A.3 Second subproblem: find $20\%$ of the total cost. Convert $20\%$ to the fraction

Review

Reasonableness: Quick sanity check: $20\%$ of $\$10$ would be $\$2$, so $20\%$ of a number a bit bigger than $\$10$ should be a bit bigger than $\$2$. The answer $\$2.50$ fits, and it sits between (B) $\$2.00$ and (D) $\$2.75$ in the right place. The two extreme choices fail: (A) $\$1.00$ would be only $8\%$ of $\$12.50$, and (E) $\$5.00$ would be $40\%$ — neither matches the $20\%$ discount.

Alternative: Tool #16 (Change Focus): instead of computing $20\%$ of the dollar total, notice that $20\% = \tfrac{1}{5}$, and Karl bought exactly $5$ folders. So $20\%$ off the purchase equals the cost of $\tfrac{1}{5} \times 5 = 1$ folder, which is $\$2.50$. Same answer (C), no multiplication of a decimal needed.

CCSS standards used (min grade 6)

5.NBT.B.7 Multiply decimals to hundredths using strategies based on place value (Computing the original total cost $5 \times \$2.50 = \$12.50$ with decimal multiplication.)
6.RP.A.3 Find a percent of a quantity as a rate per $100$ (Taking $20\%$ of $\$12.50$ to get the $\$2.50$ discount that Karl would have saved.)

⭐ $20\%$ off five folders is the same as one folder free — that shortcut turns this AMC 8 problem into a Grade 6 percent-of-a-quantity exercise.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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